Consider the ratio of sides in each golden rectangle. In Rectangle 1, the ratio is ø to 1. In Rectangle 1 + 2, the ratio is ø + 1 to ø. These are similar rectangles, so the ratios must be equal:
Cross-multiplying and simplifying the equation gives us a quadratic equation: ø2 - ø - 1 = 0. This equation does not factor, so we must use the quadratic formula to find the value of ø; the two possible values are
Since the side of a rectangle can't be negative,
Evaluating this on a calculator gives us the decimal 1.618 to three decimal places, as expected.
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